Section II. Homomorphisms 183 II Homomorphisms The definition of isomorphism has two conditions. In this section we will consider the second one. We will study maps that are required only to preserve structure, maps that are not also required to be correspondences. Experience shows that these maps are tremendously useful. For one thing we shall see in the second subsection below that while isomorphisms describe how spaces are the same, we can think of these maps as describing how spaces are alike. II.1 Definition 1.1 Definition A function between vector spaces h: V W that preserves addition ! if ~v ,~v V then h(~v + ~v )=h(~v )+h(~v ) 1 2 2 1 2 1 2 and scalar multiplication if ~v V and r R then h(r ~v)=r h(~v) 2 2 · · is a homomorphism or linear map. 1.2 Example The projection map ⇡: R3 R2 ! x ⇡ x y 0 1 7! y z ! B C @ A is a homomorphism. It preserves addition x1 x2 x1 + x2 x1 x2 x1 + x2 ⇡(0y11+0y21)=⇡(0y1 + y21)= = ⇡(0y11)+⇡(0y21) y1 + y2! z1 z2 z1 + z2 z1 z2 B C B C B C B C B C @ A @ A @ A @ A @ A and scalar multiplication. x1 rx1 x1 rx1 ⇡(r 0y11)=⇡(0ry11)= = r ⇡(0y11) · ry1! · z1 rz1 z1 B C B C B C @ A @ A @ A This is not an isomorphism since it is not one-to-one. For instance, both ~0 and 3 2 ~e3 in R map to the zero vector in R . 184 Chapter Three. Maps Between Spaces 1.3 Example The domain and codomain can be other than spaces of column vectors. Both of these are homomorphisms; the verifications are straightforward. (1) f : P P given by 1 2 ! 3 a + a x + a x2 a x +(a /2)x2 +(a /3)x3 0 1 2 7! 0 1 2 (2) f2 : M2 2 R given by ⇥ ! ab a + d cd! 7! 1.4 Example Between any two spaces there is a zero homomorphism,mapping every vector in the domain to the zero vector in the codomain. 1.5 Example These two suggest why we use the term ‘linear map’. (1) The map g: R3 R given by ! x g y 3x + 2y - 4.5z 0 1 7! z B C @ A is linear, that is, is a homomorphism. The check is easy. In contrast, the map gˆ: R3 R given by ! x g y ˆ 3x + 2y - 4.5z + 1 0 1 7! z B C @ A is not linear. To show this we need only produce a single linear combination that the map does not preserve. Here is one. 0 1 0 1 gˆ(001 + 001)=4 gˆ(001)+gˆ(001)=5 0 0 0 0 B C B C B C B C @ A @ A @ A @ A 3 2 (2) The first of these two maps t1,t2 : R R is linear while the second is ! not. x x t 5x - 2y t 5x - 2y y 1 y 2 0 1 7! x + y 0 1 7! xy z ! z ! B C B C Finding a linear@ A combination that the@ secondA map does not preserve is easy. Section II. Homomorphisms 185 So one way to think of ‘homomorphism’ is that we are generalizing ‘isomor- phism’ (by dropping the condition that the map is a correspondence), motivated by the observation that many of the properties of isomorphisms have only to do with the map’s structure-preservation property. The next two results are examples of this motivation. In the prior section we saw a proof for each that only uses preservation of addition and preservation of scalar multiplication, and therefore applies to homomorphisms. 1.6 Lemma A homomorphism sends the zero vector to the zero vector. 1.7 Lemma The following are equivalent for any map f: V W between vector ! spaces. (1) f is a homomorphism (2) f(c1 ~v1 + c2 ~v2)=c1 f(~v1)+c2 f(~v2) for any c1,c2 R and ~v1,~v2 V · · · · 2 2 (3) f(c1 ~v1 + + cn ~vn)=c1 f(~v1)+ + cn f(~vn) for any c1,...,cn R · ··· · · ··· · 2 and ~v ,...,~v V 1 n 2 1.8 Example The function f: R2 R4 given by ! x/2 x f 0 0 1 y! 7! x + y B C B 3y C B C @ A is linear since it satisfies item (2). r1(x1/2)+r2(x2/2) x1/2 x2/2 0 0 0 0 1 = r1 0 1 + r2 0 1 r1(x1 + y1)+r2(x2 + y2) x1 + y1 x2 + y2 B C B C B C B r1(3y1)+r2(3y2) C B 3y1 C B 3y2 C B C B C B C @ A @ A @ A However, some things that hold for isomorphisms fail to hold for homo- morphisms. One example is in the proof of Lemma I.2.4,whichshowsthat an isomorphism between spaces gives a correspondence between their bases. Homomorphisms do not give any such correspondence; Example 1.2 shows this and another example is the zero map between two nontrivial spaces. Instead, for homomorphisms we have a weaker but still very useful result. 1.9 Theorem A homomorphism is determined by its action on a basis: if V is a vec- tor space with basis β~ ,...,β~ ,ifW is a vector space, and if w~ ,...,w~ W h 1 ni 1 n 2 (these codomain elements need not be distinct) then there exists a homomor- phism from V to W sending each β~ i to w~ i, and that homomorphism is unique. 186 Chapter Three. Maps Between Spaces Proof For any input ~v V let its expression with respect to the basis be 2 ~v = c β~ + +c β~ . Define the associated output by using the same coordinates 1 1 ··· n n h(~v)=c w~ + + c w~ .Thisiswelldefinedbecause,withrespecttothe 1 1 ··· n n basis, the representation of each domain vector ~v is unique. This map is a homomorphism because it preserves linear combinations: where v~ = c β~ + + c β~ and v~ = d β~ + + d β~ , here is the calculation. 1 1 1 ··· n n 2 1 1 ··· n n h(r ~v + r ~v )=h((r c + r d )β~ + +(r c + r d )β~ ) 1 1 2 2 1 1 2 1 1 ··· 1 n 2 n n =(r c + r d )w~ + +(r c + r d )w~ 1 1 2 1 1 ··· 1 n 2 n n = r1h(~v1)+r2h(~v2) This map is unique because if hˆ : V W is another homomorphism satisfying ! that hˆ(β~ i)=w~ i for each i then h and hˆ have the same effect on all of the vectors in the domain. hˆ(~v)=hˆ(c β~ + + c β~ )=c hˆ(β~ )+ + c hˆ(β~ ) 1 1 ··· n n 1 1 ··· n n = c w~ + + c w~ = h(~v) 1 1 ··· n n They have the same action so they are the same function. QED 1.10 Definition Let V and W be vector spaces and let B = β~ ,...,β~ be a h 1 ni basis for V. A function defined on that basis f: B W is extended linearly ! to a function fˆ: V W if for all ~v V such that ~v = c β~ + + c β~ ,the ! 2 1 1 ··· n n action of the map is fˆ(~v)=c f(β~ )+ + c f(β~ ). 1 · 1 ··· n · n 1.11 Example If we specify a map h: R2 R2 that acts on the standard basis ! E2 in this way 1 -1 0 -4 h( )= h( )= 0! 1 ! 1! 4 ! then we have also specified the action of h on any other member of the domain. For instance, the value of h on this argument 3 1 0 1 0 5 h( )=h(3 - 2 )=3 h( ) - 2 h( )= -2! · 0! · 1! · 0! · 1! -5! is a direct consequence of the value of h on the basis vectors. Later in this chapter we shall develop a convenient scheme for computations like this one, using matrices. Section II. Homomorphisms 187 1.12 Definition A linear map from a space into itself t: V V is a linear trans- ! formation. 1.13 Remark In this book we use ‘linear transformation’ only in the case where the codomain equals the domain. However, be aware that other sources may instead use it as a synonym for ‘homomorphism’. 1.14 Example The map on R2 that projects all vectors down to the x-axis is a linear transformation. x x y! 7! 0! 1.15 Example The derivative map d/dx: P P n ! n d/dx a + a x + + a xn a + 2a x + 3a x2 + + na xn-1 0 1 ··· n 7! 1 2 3 ··· n is a linear transformation as this result from calculus shows: d(c1f + c2g)/dx = c1 (df/dx)+c2 (dg/dx). 1.16 Example The matrix transpose operation ab ac cd! 7! bd! is a linear transformation of M2 2. (Transpose is one-to-one and onto and so is ⇥ in fact an automorphism.) We finish this subsection about maps by recalling that we can linearly combine maps. For instance, for these maps from R2 to itself x f 2x x g 0 and y! 7! 3x - 2y! y! 7! 5x! the linear combination 5f - 2g is also a transformation of R2. x 5f-2g 10x y! 7! 5x - 10y! 1.17 Lemma For vector spaces V and W, the set of linear functions from V to W is itself a vector space, a subspace of the space of all functions from V to W.